The evolutions of a neutron star’s rotation and magnetic field (B-field) have remained unsolved puzzles for over half a century. We ascribe the rotational braking torques of pulsar to both components, the standard magnetic dipole radiation (MDR) and particle wind flow (MDR + Wind, hereafter named MDRW), which we apply to the Crab pulsar (B0531 + 21), the only source with a known age and long-term continuous monitoring by radio telescope. Based on the above presumed simple spin-down torques, we obtain the exact analytic solution on the rotation evolution of the Crab pulsar, together with the related outcomes as described below: (1) unlike the constant characteristic B-field suggested by the MDR model, this value for the Crab pulsar increases by a hundred times in 50 kyr while its real B-field has no change; (2) the rotational braking index evolves from ∼3 to 1 in the long-term, however, it drops from 2.51 to 2.50 in ∼45 years at the present stage, while the particle flow contributes approximately 25% of the total rotational energy loss rate; (3) strikingly, the characteristic age has the maximum limit of ∼10 kyr, meaning that it is not always a good indicator of a real age. Furthermore, we discussed the evolutionary path of the Crab pulsar from the MDR to the wind domination by comparing with the possible wind braking candidate pulsar PSR J1734-3333.
